Cascaded migrations: Improving the accuracy of finite‐difference migration

Abstract

The accuracy of time migrations done with finite‐difference schemes deteriorates with increasing reflector dip. Some properties of migration in general, and of finite‐difference approaches in particular, suggest a way of improving the accuracy of finite‐difference schemes for migrating steep dips. First, although data will be undermigrated when too low a velocity is used in migration, a correctly migrated result can be obtained by migrating again, this time with the previously undermigrated result as input. In fact, a sequence of undermigrations will yield the correct result as long as the sum of the squares of the migration velocities used in the different migration stages equals the square of the correct migration velocity. A second property is that the apparent spatial dip of a reflector perceived by the migration process is a function of not only the time dip of the unmigrated reflection, but also the velocity used in the migration. In a sequence of low‐velocity migrations, the apparent spatial dip perceived at each migration stage can be considerably less than the true dip. Thus, because finite‐difference migration is accurate for small spatial dips, the cascaded migrations yield a more accurate result than that of single‐stage migration. Also, because each migration stage is done with low velocity, the depth step can be large; hence, the computational effort need not be. The accuracy of the method is not compromised (in fact, it improves) in media in which velocity increases with depth. Moreover, the cascaded approach suffers no more than other methods of time migration where velocity varies mildly in the lateral direction. In applications of the method to stacked data from the Gulf of Mexico, reflections from near‐vertical flanks of salt domes were migrated with accuracy comparable to that achieved by frequency‐wavenumber domain migration.